Minimizing Quadratic Functions in Constant Time

A sampling-based optimization method for quadratic functions is proposed. Our method approximately solves the following n-dimensional quadratic minimization problem in constant time, which is independent of n: z∗ = minv∈Rn〈v, Av〉 + n〈v,diag(d)v〉 + n〈b,v〉, where A ∈ Rn×n is a matrix and d, b ∈ R are vectors. Our theoretical analysis specifies the number of samples k(δ, ) such that the approximated solution z satisfies |z − z∗| = O( n) with probability 1− δ. The empirical performance (accuracy and runtime) is positively confirmed by numerical experiments.